# Function Restrictions and Extensions

# Why

The relationship between the inclusion map and
the identity map is characteristic of making
small functions out of large ones.

# Definition

Let $X \subset Y$ and $f: Y \to Z$.
There is a natural function $g: X \to Z$,
namely the one defined by $g(x) = f(x)$ for
all $x \in X$.
We call $g$ the
restriction of $f$ to $X$.
We call $f$ an extension
of $g$ to $Y$.
Clearly, there may be more than one extension
of a function

## Notation

We denote the restriction of $f: Y \to Z$ to
the set $X \subset Y$ by $f\mid X$ or
$f_{\mid X}$.

## Example

A simple example is the that the inclusion
mapping from $X$ to $Y$ with $X \subset Y$ is
a restriction of the identity map on $X$

# An extension order

Here is a natural order involving set
extensions and restrictions.
Fix two sets $A$ and $B$.
Let $F$ be the set of all functions $f: X
\to Y$ with $X \subset A$ and $Y \subset B$.
Define a relation $R$ in $F$ by $(f, g) \in
R$ if $\dom f \subset \dom g$ and $f(x) =
g(x)$ for all $x$ in $\dom f$.
In other words, $(f, g) \in R$ if $f$ is a
restriction of $g$ (or, equivalently, $g$ is an
extension of $f$.
We recognize that $R$ is a special case of
the inclusion partial order by recognizing the
elements of $F$ as subsets $A \times B$.